Coordinated and controlled mobility of multiple sinks for maximizing the lifetime of wireless sensor networks

Abstract

We define scalable models and distributed heuristics for the concurrent and coordinated movement of multiple sinks in a wireless sensor network, a case that presents significant challenges compared to the widely investigated case of a single mobile sink. Our objective is that of maximizing the network lifetime defined as the time from the start of network operations till the failure of the first node. We contribute to this problem providing three new results. We first define a linear program (LP) whose solution provides a provable upper bound on the maximum lifetime possible for any given number of sinks. We then develop a centralized heuristic that runs in polynomial time given the solution to the LP. We also define a deployable distributed heuristic for coordinating the motion of multiple sinks through the network. We demonstrate the performance of the proposed heuristics via ns2-based simulations. The observed results show that our distributed heuristic achieves network lifetimes that are remarkably close to the optimum ones, resulting also in significant improvements over the cases of deploying the sinks statically, of random sink mobility and of heuristics previously proposed for restricted sink movements.

Appendix: Computing transient configurations

A transition between configuration c _i  ∈ C to configuration c _j  ∈ C is legal if $|c_{j} - c_{i}|\le s - |c_{i}|$. That is, in a legal transition, the number of sites that receive new active sinks in the new configuration does not exceed the number of inactive sinks in the preceding configuration. The inactive sinks can move to the new sites during the time configuration c _i holds. In addition we require that at least one sink is active at all times.

We now show that we can transition between any pair of major configurations using at most two transient configurations.

Theorem 1

At most two transient configurations suffice to move from any major configuration to the following one.

Proof

Suppose we wish to transition from configuration c _i  ∈ C to c _j  ∈ C where $c_i \neq c_j$. There are only three possible cases for the number of transient configurations between c _i and c _j :

• 1. $c_i \subset c_{j}$ or $c_{j} \subset c_i$. In this case, there are no transient configurations necessary since moving from c _i to c _j or vice versa satisfies our legal-transition criteria. If $c_i \subset c_j$, then $|c_j - c_i| = |c_j| - |c_i| \le s - |c_i|$, since $|c_j| \le s$. If $c_j \subset c_i,\,|c_j - c_i| = 0\,\le\,s - |c_i|$, since $|c_i| \le s$.

• 2. $c_i \not \subset c_j$ and $c_j \not \subset c_i$, and either $c_i \cap c_j \neq \emptyset$ or |c _i | < s or |c _j | < s. In this case, one transient configuration suffices. If $c_i \cap c_j \neq \emptyset$, this intersection is a legal transient configuration c ^a _i . From the argument for the previous case, we can move directly from configuration c _i to $c_i^a = c_i \cap c_j$ because $c_i \cap c_j \subset c_i$. Similarly we can move directly from configuration c ^a _i to c _j because $c_i^a = c_i \cap c_j \subset c_j$. If $c_i \cap c_j = \emptyset$ but |c _i | < s, then let c ^a _i be any element of c _j . Moving from c ^a _i to c _j is legal because $c_i^a \subset c_j$. Since |c _i | < s and configuration sizes are integers, we have $s - |c_i| \,\ge\,1$. Moving from c _i to c ^a _i is legal because $|c_i^a - c_i| = 1\,\le\,s - |c_i|$. A similar argument holds if |c _j | < s.

• 3. $c_i \not \subset c_j$ and $c_j \not \subset c_i,\,c_i \cap c_j = \emptyset$ and $|c_i| = |c_j| = s$. This covers all the remaining cases. Two transient configurations suffice. Let c ^a _i be any nonempty proper subset of c _i and let c ^b _i be any nonempty proper subset of c _j such that $|c_i^a| + |c_i^b| = s$. We can move directly from configuration c _i to c ^a _i because $c_i^a \subset c_i$. Similarly, we can move directly from c ^b _i to c _j because $c_i^b \subset c_j$. We can move directly from c ^a _i to c ^b _i because $|c_i^a - c_i^b| = |c_i^a| = s - |c_i^b|$. The first equality follows because $c_i \cap c_j = \emptyset$ and $c_i^a \subset c_i$ and $c_i^b \subset c_j$. Thus $c_i^a \cap c_i^b = \emptyset$. The second equality follows by construction.

$\square$

In the part of the proof of Theorem 1 concerning the case with one transient configuration, there were three separate sufficient conditions: $c_i \cap c_{i+1} \neq \emptyset$ or |c _i | < s or |c _i+1| < s. Our argument that one transient configuration suffices used three different constructions for that transient configuration, one for each condition. However, these conditions and constructions are independent and fully compatible. We can combine all the pieces to get a transient configuration with a maximal number of active sinks. This heuristically minimizes the total energy the transient configuration requires per unit time.

A maximal legal configuration c ^a _i for this case contains the sites in $c_i \cap c_{i+1}$ plus s − |c _i | elements of $c_{i+1} - c_i$ and s − |c _i+1| elements of $c_i - c_{i+1}$. For example, suppose $c_i = \{v_1, v_2, v_3\}$ and $c_{i+1} = \{v_3, v_4, v_5\}$, with $s\,=\,4$. Then a possible single transient state is $c_i^a = \{v_2, v_3, v_4\}$. At the start of the transition, the one spare sink, not active during configuration c _i , has finished traveling to location v ₄. To move to configuration c ^a _i , the sink waiting at site v ₄ activates while the sink at location v ₁ deactivates and begins moving to location v ₅. When the traveling sink arrives at v ₅, it activates, while the (now spare) sink at location v ₂ deactivates to put the system in configuration c _i+1.

The specific way we construct a single transient configuration between c _i and c _i+1 in step 4 of the centralized heuristic is to include the entire intersection $c_i \cap c_{i+1}$ plus s − |c _i | randomly selected sites from $c_{i+1} - c_i$ and s − |c _i+1| randomly selected sites from $c_i - c_{i+1}$. There may be some advantage to making more clever choices for the spare sinks, but this case rarely arises. In practice, if a configuration does not use all s sinks, it is almost always a pure subset of at least one other configuration.

If configurations c _i and c _i+1 have no site in common and both have size s, we compute two transient configurations as in the proof of Theorem 1. That is, we will choose c ^a _i a nonempty proper subset of c _i and c ^b _i a nonempty proper subset of c _i+1 such that $|c_i^a| + |c_i^b| = s$. For example, in Fig. 1, both configurations c _i and c _i+1 use all 6 possible sinks. $c_i^a = \{v_1, v_3, v_5\}$ and $c_i^b = \{v_7, v_9, v_{11}\}$. To make the transition, the sinks at sites in $c_i - c_i^a$ move first to locations in c ^b _i while those in c ^a _i stay where they are to receive data. Then the sinks in c ^a _i move to their final positions while those in c ^b _i receive data. There are many possible ways to choose c ^a _i as a subset of c _i and c ^b _i as a suitably-sized subset of c _i+1. We now describe one way to pick these sets and argue why heuristically we might expect it to perform well compared to using an arbitrary pair of sets that satisfy our constraints.

To create the transient configurations between major configurations c _i and c _i+1, we first consider configuration c _i . We estimate the distance between each pair of sink sites in c _i . We then construct a complete graph that has a node for each site in c _i , and edges weighted by this pairwise distance estimate. We find a minimum-weight maximum-cardinality matching in this graph. We then pick an element from each matched pair arbitrarily to form the set of sink sites c ^a _i . For instance, in the upper left corner of Fig. 1, lines show the matching for configuration c _i . The configuration $c_i^a = \{v_1, v_3, v_5\}$ is the unboxed element of each matched pair. The set of sink sites to move first is $M_1 = c_i - c_i^a = \{v_2, v_4, v_6\}$. To move from c _i to c ^a _i , the sinks at sites in M ₁ deactivate and begin to move while those in c ^a _i stay to receive traffic. We hope that if sites v _i and v _j are matched and a sink at site v _i deactivates to move, the nodes sending to v _i will be redirected to the nearby site v _j , maintaining an energy consumption pattern among nodes that is similar to that of configuration c _i . We then compute a similar matching in c _i+1 and use that to pick the new locations M ₂ for the sinks from sites in M ₁ to move to. Thus $c_i^b = M_2$. We hope c ^b _i approximates the energy balance of configuration c _i+1.